Wikipedia:Reference desk/Archives/Mathematics/2010 December 15

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December 15

Something weird

In Lang's algebra page 813 he says "Let R be a principal ring. A module is flat iff its torsion free" and he says proof is "easy". Take the ring to be Z/4Z, then its flat over itself yet is not torsion free. Infact any principal non ideal domain will do. Am I missing something? Money is tight (talk) 02:29, 15 December 2010 (UTC)

I think Lang assumes that principal rings are domains. I can't say I like the terminology. Certainly your argument is valid. --Sam^Talk 18:45, 16 December 2010 (UTC)

Laws of Mathematics

Is it possible that some of the basic laws of logic and/or math are empirical in nature? That is, we take them to be true because they are intuitively obvious, but that they are only intuitively "obvious" because the universe is structured that way.

For example, if A = B, and B = C, then A = C. Can this be proved? If so, what assertions does the proof require, and can those assertions be proved?

I suspect these questions cannot be answered, but I'm curious to hear the thoughts of those more learned in math than myself. 74.15.138.27 (talk) 09:23, 15 December 2010 (UTC)

Any useful mathematical system must start with a set of axioms - propositions that are taken to be true, and do not need to be proved. These axioms do not have to be intuitively obvious - although sometimes they are - and neither do they have to correspond to the behaviour of anything in the real world - although sometimes they do this too. Sometimes we find, by accident or by design, that the mathematical system that is built on our axioms is a useful model of some aspect of the real world. But even in this case, I don't think the axioms are empirical, because an empirical proposition can be disproved or refined by the results of an experiment. The experimental results that validated relativity and disproved Newtonian mechanics, for example, did not disprove the mathematics behind Newtonian mechanics - they just showed that this mathematical structure is not always an accurate model of reality. Gandalf61 (talk) 12:13, 15 December 2010 (UTC)

There's a whole subject Philosophy of mathematics that deals with questions like that, but there aren't any real answers. It's certainly been proposed that the laws of mathematics are empirical. The article abductive reasoning might also interest you. 67.117.130.143 (talk) 19:45, 15 December 2010 (UTC)

You might like to read the article on equivalence relations. The standard equality that we know from arithmetic is an example of an equivalence relation. In the case of the standard equality, on the set of whole numbers {…, −2, −1, 0, 1, 2, …}, it is an equivalence relation because for any whole number n we have n = n. For any two whole numbers m and n: if m = n then n = m. For any three whole numbers k, m and n: if k = m and m = n then k = n. — Fly by Night (talk) 02:34, 17 December 2010 (UTC)

Series

If sum(x_n) is a convergent series, must it be that sum((xn)^3) is a convergent series? it is obviously so if all x_n > 0 but what if there are negative terms? thanks —Preceding unsigned comment added by 131.111.222.12 (talk) 10:38, 15 December 2010 (UTC)

No. Take

$x_{n}={\begin{cases}{\frac {1}{\log n}}&3|n\\{\frac {-1}{2\log n}}&3\not |n\end{cases}}$

-- Meni Rosenfeld (talk) 13:41, 15 December 2010 (UTC)

It works if the series is absolutely convergent, i.e. the sum of the absolute values of the terms is finite. Michael Hardy (talk) 21:13, 15 December 2010 (UTC)

Or $x_{n}={\frac {(-1)^{n}}{\sqrt {n}}}$. The point being to take advantage of the fact that squaring will turn an alternating series into an all-positive series.--71.175.63.136 (talk) 16:38, 15 December 2010 (UTC)

And I misread that exponent as 2 instead of 3. Ignore me.--71.175.63.136 (talk) 16:39, 15 December 2010 (UTC)

centre of circle

Three points A(a,b) B(c,d) ,D(e,f) are three points on circle in a plane .is there any simple formula to find their centre (h,k) ,I have derived one ,but i think perhaps it will also be derived earlier .Is their short formula .how can i do research in a field of mathematics such that same can not be done in Mathematics. khan — Preceding unsigned comment added by True path finder (talk • contribs) 12:00, 15 December 2010 (UTC)

See if your formula agrees with the one given here: http://mathforum.org/library/drmath/view/55239.html. If you just want to check examples, Wolfram Alpha can do it: http://www.wolframalpha.com/input/?i=circle+through+%281%2C2%29+%280%2C0%29+%283%2C0%29.

Is your more general question about finding a research field where your work is unlikely to be already done by somebody else? If that's what you want to do, the traditional way is to work your way through a mathematics graduate school program, and build relationships with good mentors who can help you find open problems in their field which you could realistically work on. It is still possible for people outside of this traditional system to do new research in mathematics, but this is extremely rare. Staecker (talk) 13:40, 15 December 2010 (UTC)

There is a method from elementary geometry that you might be interested it. Connect the three points A, B, and C into a triangle. The circumcentre of the triangle will be the centre of the circle in which the triangle is inscribed. It is a trivial exercise of algebra to compute the perpendicular bisectors and their point of intersect. 24.92.70.160 (talk) 00:44, 16 December 2010 (UTC)